EXPANSION OF ENTIRE FUNCTIONS OF SEVERAL COMPLEX … · 2018. 11. 16. · EXPANSION OF ENTIRE FUNCTIONS OF SEVERAL COMPLEX VARIABLES HAVING FINITE GROWTH BY P. K. KAMTHAN AND MANJUL

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 192, 1974

EXPANSION OF ENTIRE FUNCTIONS OF SEVERAL COMPLEXVARIABLES HAVING FINITE GROWTH

BYP. K. KAMTHAN AND MANJUL GUPTA(l)

ABSTRACT. We consider the space x of entire functions of two complex variableshaving a finite nonzero order point and type, equip it with the natural locally convextopology, such that x becomes a Frechet space. Apart from finding the characterization ofcontinuous linear functionals, linear transformations on x, we have obtained the necessaryand sufficient conditions for a double sequence in x to be a proper base.

1. Introduction and terminology. Let x denote the space of all entire functionsof two variables zx, z2 E C (C is the ordinary complex plane equipped with itsusual topology) endowed with the topology $ of uniform convergence oncompact sets in C2. The topology 'f can also be regarded as the topologygenerated by the family [M(f; RX,R2): RX,R2*> 0}, where

M(f,rx,r2)= sup |/f>„z2)l.

and hence 5"can also be expressed in terms of arithmetical mean values off; butcertainly not in terms of geometric mean values of/(for counterexample, see [5]).In an earlier paper [4] one of us has made a systematic study of the topologicalstructure of the space x leading to some characterizations of linear functionals,operators and proper bases in x- In all these results, functions of finite orders andtypes nowhere occupy a privileged position, although from the point of view ofclassical analysis, such functions are of immense importance. Our aim is,therefore, to consider the family of such class of entire functions of two variables(for the sake of simplicity, we consider the case of two variables only) and makea topological study thereon. Various notions of orders and types regardingfunctions in x are available nowadays (see for instance the book by Fuks [2] andreferences given therein). We, however, follow those mentioned by Fred Gross[3]. Indeed, Fred Gross has given characterization of orders and types in termsof the coefficients of the expansion of an entire function/ E x, and this very wellfits in our analysis in a most general setting.

For simplification, it is sometimes convenient to introduce various equivalentlocally convex topologies on x apart from f mentioned earlier. For instance, let

Received by the editors August 3, 1972 and, in revised form, February 27, 1973.AMS (MOS) subject classifications (1970). Primary 32A15; Secondary 46A05, 46A35.Key words and phrases. Entire functions, order, type, mean values, locally convex topology,

bases, linear functionals, transformations, proper bases.(') The research work of this author has been supported by the Junior Fellowship of CSIR,

Government of India, and a junior research assistantship, MX, Kanpur.Copyright O 1974, American Mathematical Society

371

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

372 P. K. KAMTHAN AND MANJUL GUPTA

for each / E x

(1.1) 11/11 = supflflool,km\V(^"\m,n > 0,m + n # 0).For Rx, R2 > 0, let

0-2) p(/;P„P2) = 2 2 kJ*r*5,n=0 m=0

where

0-3) fax,**) =55 a^zl, amn E C,B-0 m=0

with |amB|1,/(m+'') -» 0 as m + n -* oo. Then ||/|| is a total paranorm andp(/; RX,R2) is a norm on x- Let T and T2 be the topologies generated by ||/|| andthe family [p(f;Rx,R2);Rx,R2 > 0} respectively. It is easy to see that 'S ̂ -Txa* T2. Indeed, let {fp} be a net in x, such that^ -»/in T^. Choose e > 0, tj > 0,P,, P2 > 0 arbitrarily, such that

t,P„ 1,/ï, < 1, i, + ■r\2RxR2/{(\ - i»*i)0 - iIÄ2)} < e.

Now for p > Q = Q(r¡)

1^-flboKl. tó)-amB| 0, we may findP„ P2 > 0 with P„ P2 > 1/e and a ß = g(e,P,,P2), such that

1^-flboK«. forp>ß;

U(*i>*2) -/(^i>^)l < 1. forp > Q and |z,| < P„ i = 1, 2.Using Cauchy's fundamental inequality for two variables [2, p. 49], namely,\amn\R\R2 < M(f; RX,R2), one finds for m + n * 0,

Wtí - aJ1/(",+") < Rx-m^m+n)R2"^m+n) < 1/P < e, forp > Q,

where P = min(RX,R2) > 1/e. Thusj£ -+finT. We have, therefore, shown thatS as T. The other part, namely 'S at T2, follows from the following inequalities:

(1.4) M(f;rx,r2) < p(f;rx,r2) < -¡^L- ' ffi-:M(f;Rx,R2),K\ 1 K2 r2where R¡ > r¡, > 0 (i = 1,2). Inequalities (1.4) follow from the Cauchy inequal-ity referred to above.


ENTIRE FUNCTIONS OF SEVERAL COMPLEX VARIABLES 373

We now mention below the result of Fred Gross [3] on which our analysisdepends. Let px, p2 ; ox, o2 be four positive finite numbers which are arbitrary butfixed throughout our discussion. Let Y = Y(px,p2;ox,o2) be the subclass of xconsisting of those functions / such that for each e > 0

(1.5) M(f;rx,r2) < exp{(o-, + eV + (o2 + e)r?),

for all sufficiently large values of rx and r2. Clearly y is a proper vector subspaceof x- Since each/ e Y is represen table as given in (1.3), one has after Fred Grossthe following:

(..6) k.i < [£a(a¡±i)]*[a2k±i]"»,

valid for all sufficiently large m + « and conversely if / G x and satisfies (1.6),then/satisfies (1.5), i.e./e Y.

To consider the topological structure of Y, define for each f E Y and each8 > 0, the following term:

||/;o,+Ó\a2 + S||

= hol + 2, 2, kJ [j^Ts^X* [(o-Told"7*'where the double infinite series converges in view of (1.6). Then for each 5 > 0,||/; ox + 8,o2 + 8\\ defines a norm on Y. Let S be the topology on Y generated bythe family {||/;oi + 8,o2 + 8\\: 8 > 0}. We will throughout assume from nowand onwards, unless otherwise stated, that Y is equipped with the topology S. Itis then easily seen that Y is a locally convex space. Moreover, it is metrizable, themetric being given by

j(fo\ = v l H/-g;qi + iA>*2 + iAIIaU'8) k% 2* 1 + ||/- g; a, + \/k,o2 + \/k\\ •

Moreover, it can be easily verified that the topology induced on Y by ?F is weakerthan the original topology on Y generated by d. In x, the sequence of partial sumsof /= 2r=o2"=o«/wi5m«. Smn(zx,z2) = zfzl converges to /. This does notfollow easily in the case of Y when it is equipped with the topology 9. We showthat in fact it is true.

Theorem 1.1. Let f = 2"=o 2"=o amn^mn G Y. The sequence of partial sums ofthe series for fconverges to fin Y with respect to S.

Proof. If we prove that the assertion of the theorem is true in Y(8) for every8 > 0, where Y(8) the vector space Y equipped with the norm (1.7), then theresult will follow. Let 8 > 0 be given. In view of (1.6), we can choose rj < 8 andAT = Ar(n) such that



kJ N(t\).If SN is the partial sum of the series for/, SN = Wa


Characterization of continuous linear functionals on Y. First, we prove thefollowing theorem:

Theorem 2.2. A continuous linear functional on Y(8) is of the form

(2-1)


3. Continuous linear transformations of Y into itself. Let a continuous lineartransformation from Y(8X) into Y(82) be denoted by T(8X,82) and the family ofall such transformations for a fixed pair 5, and 82 by B(8X,82). A continuous lineartransformation from Y into itself is denoted by T and the family of suchtransformations is B(Y). The main result of this section is stated as

Theorem 3.1. A linear transformation T E B(Y) if and only if for each 8 > 0there exists a constant k = k(8) and an e, = e,(S) such that

(3.1) ||r(0;*i + «,»2 + «II < k\, "\ T/Pl[, /VT*L (a, + e, )ep, J |_ (a/nes of m,n > 0.

For proving this result, we shall need the following lemmas:

Lemma 3.1. If for a E Y, d(a) > k > 0, then the following must be true:

||«; o, + 8,a2 + ¿|| > Jfc/(2 - k)

for some 8 = 80, where 0 < ô0 < 1, and therefore for all values of8< 80.

Proof. Consider the sequence of norms ||a;a, + l/p,a2 + l/p||. This increasesfor increasing values of p. Choose pQ such that

- 1 ||a;a,+ l/p,a2+l/p|| k^,2M + ||«;a, + l/p,a2+l/pll 2'

Then we have

j(„\ sk ± ||a;g, + l/po,g2+l/p0|| ÇI 1 J_\l"K2 1 + II«; o, + l/p0,o2 + l/p0|| \2 + 22 + ' ' ' + 2p°j

^M \ ^ II«? Q| + 1/P0,g2+ 1/Poll , kw^ 1 +||a;a, + l/po,a2+l/pol| 2

Sincera) > &, it implies \W',ox + l/p0,o2 + l/p0\\ > k/(2 - k). With50 = l/Po>the result follows.

Remark. From this lemma, it immediately follows that the convergence of aseries in Y(8) for each 5 > 0 implies its convergence in Y. Combining this resultwith the fact that the topology on Y is stronger than all the topologies on theY(8)'s, we can say that the convergence in y is equivalent to convergence in allof the y(5)'s.

Lemma 3.2. A linear transformation TofY into itself is continuous if and only ifto each 82 > 0 there exists some 5, > 0 such that T E B(8X, 82).

Proof. Since the topology on Y is stronger than all the topologies on y(5)'s, itfollows that any T E B(Y) is a continuous linear transformation from Y intoy(52) for each 52 > 0. Choose 52 arbitrarily and fix it. Therefore, for the



necessary part we need to show the existence of some 8X > 0 such thatT G B(8X,82). Let us assume that a linear transformation F of Y(8X) into Y(82)is not continuous for any 8X > 0. Then by known properties of normed spaces[1, p. 54], we can find a sequence [Y(\/p)} and ap in Y(\/p) such thatlla^îo-, + l/p,o2 + \/p\\ -* 0 as p -* oo, while \\T(ap);ox + 82,o2 + 82\\ > 1, forall sufficiently large p. This implies that d(ap) -* 0 as p -* oo; but \\T(ap);ax+ 82, o2 + 821| > 1 => T is not a continuous linear map from Y into Y(82). SinceS2 is arbitrary, it follows from the above arguments that for T G B(Y) and forany 52 > 0, we should be able to find some 8X > 0 such that T E B(SX,82).

To prove the converse, suppose that a linear transformation F of y into Y isnot continuous. Then there exists a sequence {ap} of elements of Y such thatd(ap) -> 0 as p -> oo but d(T(ap)) > k > 0, p = 1, 2,_By Lemma 3.1, wehave

||T(ap); a, + 8,o2 + ô|| > */(2 - k) for all 5 < 80, 0 < 50 < 1-

This implies that Fis not a continuous linear map from Y into Y(8) for all 8 < 80.From here, it can be easily shown that for S's < 80, we cannot find any 8X > 0such that T E B(8X,8). This completes the proof.

Proof of Theorem 3.1. Let F G B(Y) with T(8m) = a^. Then, from Lemma3.2, it follows that for every 8 > 0, there exists 8X = 5](ô) such that F is acontinuous linear map from Y(8X) to Y(8).

There exists a constant A: = k(8) such that

lin*«.);»! + S,°2 + «II < *ll««,;a, + «1.0-2 + «ill[,*, Tm/pir » T"/p2

(a, + S, )epx J L ( 0 and for all sufficiently large m + n.For fixed e > 0, there exists N0 = Aê) such that

o.« k.| Aê). From (3.1) it follows that we can find e! > e such that

(3.3) k..,+8,o2+«y < ,r^^r/ftr^Lîfl/ftL (o\ + ei )epx J L (o2 + ex )ep2 J



for all m, n > 0. Combining (3.2) and (3.3) we get

for (m + n) > N0(e)00 00

=» 2 2 Km I Ha»«; 0m-0 m-0

00 00

=* 2 2 0.For the characterization of proper bases in terms of the growth conditions on

{amn}> we lùst prove the following two lemmas:

Lemma 4.1. For a sequence {aTO} C Y, the following three conditions areequivalent:

(A) For each 5 > 0, there exists k = k(o) and an e, = c,(5) such that

\\am;ol + 8,a2 + ô\\ 0).



(B) For all sequences {cm} of complex numbers, "2n% 2"-o cna^mn converges inY" implies "2«°-o 2™«o Cmn*mn converges in Y".

(C) For all sequences {c^} of complex numbers, "2^«o 2m-o cim^mn converges inY" implies " câ^ tends to zero in Y".

Proof. From the sufficiency part of the proof of Theorem 3.1, (A) =» (B) isobvious. (B) => (C) is also quite obvious. So, in order to complete the proof, weneed show that (C) => (A). For this, suppose that (C) is true, but (A) is not true.Then the latter implies that for some 8, say 8', and every integer k there existsequences {mk} and {«*} of integers such that

■W> K.„ ♦ r„♦ „I * ,[çri%A]-[KTfe5]-.Define a sequence {cm} of scalars by

.. ,, c»« = ■■ .„ ,«,„ , « .i when m = mk;n = nk, k = 1, 2.(4.3) \\ 0, and for sufficiently large A:

for any e > 0, and for sufficiently large m and n.

Thus, c„„ defined by (4.3) satisfies (4.1) and consequently câ^ should tend tozero in F by the hypothesis (C). But for n = nk and m = mk,

Ik™«™;*, + 8',a2 + «II = \cmtnk\ \\ami¡nt;ox + 5',a2 + 5'|| = 1

which contradicts that câ^ -* 0 in Y. This establishes that (C) ** (A).

Lemma 4.2. For a sequence {am} in Y, the following conditions are equivalent:(a) For 17 > 0 and all sufficiently small Ô > 0, S = 8(rj),



ik.;*+«,a2+oh > \ >"]*«[—"T",""' ' '2 - L(a, +ij)ep,J l(o2 + 7,)ep2]where m + n is sufficiently large and depends upon 8.

(ß) For all sequences {cm} of complex numbers, "2fí.o 2m-o cmnamn converges inI"' => - 2r-o 2^=0 ^«m» converges in Y".

(y) For all sequences {c^,} of complex numbers, "Cmnamn tends to zero in Y" =>"2»°-o 2m-o cnm8mn converges in V.

Proof, (y) => 08) is obvious. We shall prove that (ß) ■♦ (a) and (a) => (y).For proving (ß) => (a), we assume that (ß) is true and (a) is not true. Then the

latter implies that for some tj > 0, for arbitrary small 80 and for arbitrarily largem + n, depending on 8q,

[m lm/pir « ~\n/PihtäI Unud •Since Ha^,; a, + 8,o2 + 5|| decreases as 5 increases, it follows from here that

(4.5, K.;, + s,, ♦ S|| < [j-^-^tç-^j-J*is true for each 5 > 0 and arbitrarily large m + n which depend on 8. If tj is afixed small positive number, we can find, corresponding to each number k > 0,increasing sequences {mk} and {nk} such that

(4.6)m*/pi r «, nVfti.! . i m _- r mk T^r "k i


arbitrary, the series 2 2m^>o câ^,, when {c„„} is defined by (4.7), is convergentin Y and consequently by our hypothesis (ß), {c^} should satisfy the condition(4.1). On the contrary, from the definition of the sequence {c^,}, it is obvious that{cmn} does not satisfy the condition (4.1). Hence our assumption is wrong and (/?)■»(«)•

To prove (a) => (y), we take a sequence {c^,} of scalars for which câ^ -* 0in Y and the condition (4.1) does not hold. Then there exist sequences {mk} and{nk} of positive integers and a number X > 0 such that

(«i ic,..i > [ssfciîïpfafetii]-'.Choose a positive number tj such that A > (3tj/2). Then from (a), we can find a8 = S(ij) such that

m ^;„1+ä,„1+i|la[;_^_]-[_^_]-

for sufficiently large m + « which depends on 8 and therefore on tj. Thereforethere exists a number N(rj) such that

ikî+o^+oii > \^—\mp2J

for m + n > N(tj).

Now, consider

max Ik™ a™;», + 8,o2 + ô||= maxkJ Ho^ja, + 5,ct2 + S||

> max|cmtnJ ||amtBt;a, + 5,ct2 + fi||

rgPl(q,+A)-|"*/". x rgp2(a2 + A)1"^

x r ™* t*7"1 x r ** t*^L(a, + ■q)epx J |_(2 + i)>P2J

>1.This implies that câ^ does not tend to zero in Y(8) corresponding to the abovechoice of 8, and this contradicts our assumption. Hence (a) => (y).

Combining the above two lemmas, we get a characterization of proper basesin the form of



Theorem 4.1. A base {am} in a closed subspace Y0of Y is proper if and only if{

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EXPANSION OF ENTIRE FUNCTIONS OF SEVERAL COMPLEX … · 2018. 11. 16. · EXPANSION OF ENTIRE FUNCTIONS OF SEVERAL COMPLEX VARIABLES HAVING FINITE GROWTH BY P. K. KAMTHAN AND MANJUL